| name | upper case | lower case | usage |
|---|---|---|---|
| Lambda | \(\Lambda\) | \(\lambda\) | Loading of a manifest indicator onto a latent construct |
| Psi | \(\Psi\) | \(\psi\) | residual variance/covariance of contruct when endogenous |
| Theta | \(\Theta\) | \(\theta\) | residual variance/covariance of indicators |
| Sigma | \(\Sigma\) | \(\sigma\) | \(\Sigma\) is the model implied variance/covariance matrix; |
| \(\sigma\) is standard deviation, \(\sigma^2\) variance of indicator. | |||
| \(\sigma\) can also be covariance of indicator |
\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{1,2} \\ \lambda_{2,1} & \lambda_{2,2} \\ \lambda_{3,1} & \lambda_{3,2} \end{array} \right],\]
\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} & \psi_{1,2} \\ \psi_{2,1} & \psi_{2,2} \end{array} \right],\]
\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} \\ \lambda_{1,2} & \lambda_{2,2} & \lambda_{3,2} \end{array} \right],\]
\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 & 0 & 0 & 0 \\ 0 & \theta_{2,2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \theta_{3,3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_{4,4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta_{5,5} & 0 \\ 0 & 0 & 0 & 0 & 0 & \theta_{6,6} \end{array} \right].\]
\[ \Sigma = \Lambda \Psi \Lambda' + \Theta \tag{1} \]
library(lavaan)
##Prepare data with sufficient statisitics##
mymeans<-matrix(c(3.06893, 2.92590, 3.11013), ncol=3,nrow=1)
mysd<-c(0.84194,0.88934,0.83470)
mat <- c(1.00000,
0.55226, 1.00000,
0.56256, 0.66307, 1.00000)
mycor <- getCov(mat, lower = TRUE)
##Transform correlation matrix to covariance matrix using information above##
mycov <- mysd %*% t(mysd)
rownames(mycor) <-c( "Glad", "Cheerful", "Happy")
colnames(mycor) <-c( "Glad", "Cheerful", "Happy")
rownames(mycov) <-c( "Glad", "Cheerful", "Happy")
colnames(mycov) <-c( "Glad", "Cheerful", "Happy")
mynob<-823
| Glad | Cheerful | Happy | |
|---|---|---|---|
| Glad | 1.00 | 0.55 | 0.56 |
Found more than one class "Model" in cache; using the first, from namespace 'MatrixModels'
# Mplus file
l.cheer.inp
using correlations only (instead of variance/covariance matirx)
lavaan (0.5-20) converged normally after 9 iterations
Number of observations 823
Estimator ML
Minimum Function Test Statistic 0.000
Degrees of freedom 0
Model test baseline model:
Minimum Function Test Statistic 0.000
Degrees of freedom 0
P-value NA
User model versus baseline model:
Comparative Fit Index (CFI) 1.000
Tucker-Lewis Index (TLI) 1.000
Loglikelihood and Information Criteria:
Loglikelihood user model (H0) -1070.768
Loglikelihood unrestricted model (H1) -1070.768
Number of free parameters 1
Akaike (AIC) 2143.536
Bayesian (BIC) 2148.249
Sample-size adjusted Bayesian (BIC) 2145.074
Root Mean Square Error of Approximation:
RMSEA 0.000
90 Percent Confidence Interval 0.000 0.000
P-value RMSEA <= 0.05 1.000
Standardized Root Mean Square Residual:
SRMR 0.000
Parameter Estimates:
Information Expected
Standard Errors Standard
Latent Variables:
Estimate Std.Err Z-value P(>|z|)
Positive =~
Cheerful 1.000
Variances:
Estimate Std.Err Z-value P(>|z|)
Positive 0.790 0.039 20.285 0.000
Cheerful 0.000
cat(file = 'topics/2_MeasurementModel/2b_ConfirmatoryFactorAnalysis/mplus/l.cheer.out')
Warning in lav_samplestats_from_moments(sample.cov = sample.cov,
sample.mean = sample.mean, : lavaan WARNING: sample covariance can not be
inverted
lavaan (0.5-20) converged normally after 116 iterations
Number of observations 823
Estimator ML
Minimum Function Test Statistic 0.000
Degrees of freedom 0
Parameter Estimates:
Information Expected
Standard Errors Standard
Latent Variables:
Estimate Std.Err Z-value P(>|z|)
Positive =~
Glad 0.841 0.021 40.570 0.000
Cheerful 0.889 0.022 40.570 0.000
Happy 0.834 0.021 40.570 0.000
Variances:
Estimate Std.Err Z-value P(>|z|)
Glad 0.000 0.000 13.060 0.000
Cheerful 0.000 0.000 12.213 0.000
Happy 0.000 0.000 13.191 0.000
Positive 1.000
\[ \textbf{$\Sigma$} = \left[ \begin{array}{cccccc}
\sigma_{1,1}^2 & \sigma_{1,2} & \sigma_{1,3} & \sigma_{1,4} & \sigma_{1,5} & \sigma_{1,6} \\
\sigma_{2,1} & \sigma_{2,2}^2 & \sigma_{2,3} & \sigma_{2,4} & \sigma_{2,5} & \sigma_{2,6} \\
\sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3}^2 & \sigma_{3,4} & \sigma_{3,5} & \sigma_{3,6} \\
\sigma_{4,1} & \sigma_{4,2} & \sigma_{4,3} & \sigma_{4,4}^2 & \sigma_{4,5} & \sigma_{4,6} \\
\sigma_{5,1} & \sigma_{5,2} & \sigma_{5,3} & \sigma_{5,4} & \sigma_{5,5}^2 & \sigma_{5,6} \\
\sigma_{6,1} & \sigma_{6,2} & \sigma_{6,3} & \sigma_{6,4} & \sigma_{6,5} & \sigma_{6,6}^2
\end{array} \right],\]
\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{1,2} \\ \lambda_{2,1} & \lambda_{2,2} \\ \lambda_{3,1} & \lambda_{3,2} \end{array} \right],\]
\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} & \psi_{1,2} \\ \psi_{2,1} & \psi_{2,2} \end{array} \right],\]
\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} \\ \lambda_{1,2} & \lambda_{2,2} & \lambda_{3,2} \end{array} \right],\]
\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 & 0 & 0 & 0 \\ 0 & \theta_{2,2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \theta_{3,3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_{4,4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta_{5,5} & 0 \\ 0 & 0 & 0 & 0 & 0 & \theta_{6,6} \end{array} \right].\]